Factor completely: 48x^4y^2+243y^2

In order to factor this completely, we need to remove all common items.

The first thing to do is to recognize that y^2 is part of both addends, and can be removed.  This gives us:
y^2(48x^4 + 243)

Now x is only in one addend, so that can't be factored out.  The only thing left is to see if the numbers can be simplified.  We can look immediately and see that 243 is not divisible by 2, so simplifying by taking out 2 is not an option.  However, we might be able to divide by a different number.  

If you can add up the numbers of a number and get a number divisible by three, then your number is divisible by three.
For example:
2+4+3=9, which is divisible by three.  That means that 243 is also divisible by three.
4+8=12, which is divisible by three.  That means 48 is also divisible by three.

We have just found out that both 48 and 243 can be divided by three, so 3 can be factored out.
This gives us:
3y^2(16x^4+81)

Please don't just take the answer I've given you and move on.  Please try to read and understand my explanation, because while it is easy to cheat on homework, you can't cheat on quizzes and tests, so you need to try to understand the process.  If anything that I wrote was confusing, just leave a comment and I will explain it as soon as I can.  :)